In this article we give two new characteristics of quasi-antiorder relation on ordered set under antiorder.\\ The new results in this article is co-called the second isomorphism theorem on ordered sets under antiorders: Let $(X,=,\neq,\alpha)$ be an ordered set under antiorder $\alpha$, $\rho$ and $\sigma$ quasi-antiorders on $X$ such that $\sigma\subseteq\rho$. Then the relation $\sigma/\rho$, defined by $$\sigma/\rho=\{(x(\rho\cup\rho^{-1}),y(\rho\cup\rho^{-1})\in X/(\rho\cup\rho^{-1})\times X/(\rho\cup\rho^{-1}):(x,y)\in\sigma\},$$ is a quasi-antiorder on $X/(\rho\cup\rho^{-1})$ and $(X/(\rho\cup\rho^{-1}))/((\sigma/\rho)\cup(\sigma/\rho)^{-1})\cong X/(\sigma\cup\sigma^{-1})$ holds.\\ Let $\mathbf{A}=\{\tau: \tau$ is quasi-antiorder on $X$ such that $\tau\subset\sigma\}$. Let $\mathbf{B}$ be the family of all quasi-antiorder on $X/q$, where $q=\sigma\cup\sigma^{-1}$. We shall give connection between families $\mathbf{A}$ and $\mathbf{B}$. For $\tau\in\mathbf{A}$, we define a relation $\psi(\tau)=\{(aq,bq)\in X/q\times X/q:(a,b)\in\tau\}$. The mapping $\psi:\mathbf{A}\rightarrow\mathbf{B}$ is strongly extensional, injective and surjective mapping from $\mathbf{A}$ onto $\mathbf{B}$ and for $\tau, \mu\in\mathbf{A}$ we have $\tau\subseteq\mu$ if and only if $\psi(\tau)\subseteq\psi(\mu)$.